There exist no locally symmetric Finsler spaces of positive or negative flag curvature

TL;DR

This paper proves that locally symmetric Finsler spaces cannot have positive or negative flag curvature, extending previous results to a local setting and clarifying the nonexistence of such metrics.

Contribution

It demonstrates that the nonexistence results for globally symmetric Finsler spaces with certain curvatures are also valid locally, refining prior global findings.

Findings

Locally symmetric Finsler spaces do not admit positive or negative flag curvature.

The nonexistence results are valid in a local context, not just globally.

Extends previous global nonexistence theorems to local cases.

Abstract

We show that the results of Foulon (1997 an 2002) and Kim (2007) (independently, Deng and Hou (2007)) about the nonexistence of locally symmetric Finsler metrics of positive or negative flag curvature are in fact local.